Practical Approximate Analysis of Beams and Frames: Lecture Notes in Mechanics 1

391 Pages·2012·5.78 MB·English

by  Nabil Fares

LectureNotesinMechanics1
Practical
Approximate
Analysis of Beams
and Frames
Nabil Fares, Ph.D.
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Lecture Notes in Mechanics
AristotleandArchytasdefinedmechanicsasthe“organizationofthought
towardssolvingperplexingproblemsthatareusefultohumanity.”Inthespirit
Aristotle and Archytas, Lecture Notes in Mechanics (LNMech) tran-
scends the traditional division of mechanics and provides a forum for
presentingstate-of-the-artresearchthattacklestherangeofcomplexis-
suesfacingsocietytoday.
LNMech provides for the rapid dissemination of comprehensive
treatmentsofcurrentdevelopmentsinmechanics,servingasarepository
andreferenceforinnovationinmechanics,acrossallrelevantapplication
domains.
LNMech publishes original contributions, including monographs,
extended surveys, and collected papers from workshops and confer-
ences. All LNMech volumes are peer reviewed, available in print and
onlinethroughASCE,andindexedbythemajorscientificindexingand
abstractingservices.
SeriesEditor
RogerGhanem
EditorialBoard
YounaneAbousleiman,Ph.D.,UniversityofOklahoma
RobertoBallarini,Ph.D.,P.E.,UniversityofMinnesota
RonaldoI.Borja,Ph.D.,StanfordUniversity
ShiyiChen,Ph.D.,PekingUniversity
TerryFriesz,Ph.D.,PennsylvaniaStateUniversity
BojanB.Guzina,Ph.D.,UniversityofMinnesota
IoannisKevrekidis,Ph.D.,PrincetonUniversity
MohammadA.Khaleel,P.E.,Ph.D.,PacificNorthwestNational
Laboratory
PeterKuhn,Ph.D.,UniversityofCaliforniaatSanDiego
ArifMasud,Ph.D.,UniversityofIllinois,Urbana-Champaign
IgorMezic,Ph.D.,UniversityofCaliforniaatSantaBarbara
KaramSab,Ph.D.,E´coleNormaledesPontsetChausse´es
AndrewW.Smyth,Ph.D.,ColumbiaUniversity
ChristianSoize,Ph.D.,Universite´ Paris-Est
PolD.SpanosPh.D.,P.E.,RiceUniversity
LizhiSun,Ph.D.,UniversityofCaliforniaatIrvine
ZhigangSuo,Ph.D.,HarvardUniversity
PaulTorrens,Ph.D.,UniversityofMaryand
Franz-JosefUlm,Ph.D.,P.E.,MassachusettsInstituteofTechnology
LNMech1 PracticalApproximateAnalysisofBeamsandFrames
NabilFares,Ph.D.
LNMech2 StochasticModelsofUncertaintiesinComputational
Mechanics
ChristianSoize,Ph.D.
Contents
Preface vii
1 Approximate Analysis of Beams and Frames with
no Sidesway 1
1.1 IntroductiontoSketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 PassiveMembersinContinuousBeamsandFrames . . . . . . . . . . . 4
1.3 BeamwithaMomentAppliedatOneEndandResisting
attheOther . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Example:ContinuousBeamwithMomentApplied
atOnlyOneNode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 OutlineofApproximateMethodforAnalyzingStructures
withNoSidesway. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 BeamwithaUniformLoad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Example:UniformLoad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.8 BeamwithaPointForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.9 Example:PointForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1.10 CommentsandExamplesonMultipleLoads. . . . . . . . . . . . . . . . . 71
1.11 BeamwithTwoorMoreInternalHinges. . . . . . . . . . . . . . . . . . . . 79
1.12 BeamwithOneInternalHinge,aMomentAppliedatOneEnd
andResistingattheOther . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.13 BeamwithOneInternalHingeandaUniformLoad. . . . . . . . . . . . . 94
1.14 BeamwithOneInternalHingeandaPointForce . . . . . . . . . . . . . . 98
2 Approximate Analysis of Frames with Sidesway 105
2.1 TheCantileverandtheSingleFloorPortalFrame. . . . . . . . . . . . . . 105
2.2 ApproximateAnalysisofSingleFloorFramesSubject
toaHorizontalLoad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.3 SketchingSingleFloorPortalFrames . . . . . . . . . . . . . . . . . . . . . 124
2.4 TheColumnwithRotarySpringsandMomentsatBothEnds . . . . . . 131
2.5 ApproximateAnalysisofMultipleFloorFramesSubject
toHorizontalLoads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.6 ANoteontheLumpedMassModelforBuildings . . . . . . . . . . . . . . 152
2.7 SketchingMultipleFloorFramesSubjecttoHorizontalLoads . . . . . . 160
2.8 NotesonSideswayDuetoVerticalLoadsorAppliedCouples . . . . . . 171
v
vi PracticalApproximateAnalysisofBeamsandFrames
3 Estimating Displacements in Beams and Frames 185
3.1 MaximumVerticalDisplacementsinBeams . . . . . . . . . . . . . . . . . 185
3.2 EstimatingMomentofInertia . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.3 RelativeVerticalDisplacementsversusStraininBeams. . . . . . . . . . 203
3.4 SideDisplacementsofFramesSubjecttoSideLoads . . . . . . . . . . . 211
3.5 ObtainingRotaryStiffnessFactorsfromSlopeMeasurements
inBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4 Approximate Influence Lines for Indeterminate
Beams 235
4.1 IntroductiontoInfluenceLines. . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.2 ExactInfluenceLinesforStaticallyDeterminateBeams . . . . . . . . . . 242
4.3 ApproximateInfluenceLinesforStatically
IndeterminateStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Appendixes
A Beams—End-Moments and Inflection Points 317
A.1 MomentEnd-LoadedBeam . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
A.2 UniformlyDistributedLoad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
A.3 PointForce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
B Column—Shear Stiffness, End-Moments and
Inflection Points 327
B.1 Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
B.2 ColumnforSingleStoryBuilding . . . . . . . . . . . . . . . . . . . . . . . . 328
B.3 ColumnforMulti-StoryBuilding—FirstFloor . . . . . . . . . . . . . . . . . 330
B.4 ColumnforMulti-StoryBuilding—TopFloor(TopandBottom
BeamsSimilar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
B.5 ColumnforMulti-StoryBuilding—GeneralCase. . . . . . . . . . . . . . . 333
C Beams—Deflections and Rotations 335
C.1 DisplacementsatAnyLocation . . . . . . . . . . . . . . . . . . . . . . . . . 335
C.2 RotationsatAnyLocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
C.3 UniformLoad—MidDisplacements. . . . . . . . . . . . . . . . . . . . . . . 340
C.4 PointForce—CentrallyLoaded—MidDisplacements. . . . . . . . . . . . 342
C.5 Pointforce—LoadedAnywhere—MidDisplacements . . . . . . . . . . . 344
C.6 PointMoment—LoadedAnywhere—MidDisplacements . . . . . . . . . 347
C.7 Cantilever—VariousSpecialCases—Displacements . . . . . . . . . . . . 350
PracticalApproximateAnalysisofBeamsandFrames vii
D Useful Results for Influence Lines 353
D.1 InfluenceLinesforVerticalForceReactions . . . . . . . . . . . . . . . . . 353
D.2 InfluenceLinesforShearForces . . . . . . . . . . . . . . . . . . . . . . . . 355
D.3 InfluenceLinesforBendingMoments . . . . . . . . . . . . . . . . . . . . . 355
D.4 TransitionMember . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Index 359
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Preface
Theaimofthisbookistopresentanewapproachtoapproximatelyan-
alyzebeamsandframes.Thenewapproachhasthefollowingdesirable
features:
• The approach is relatively short and simple, robust with good accu-
racy and is practically applicable to realistic problems. Some former
students who have learned these methods have reported that, in the
workplace,theyhavebeenabletocompleteanaccurateapproximate
analysis of a structure in the time it took their colleagues to enter a
descriptionofthestructureintoacomputermodel.
• Theapproachisnaturallyamenabletoparametricstudiesandresults
presenting summaries and ranges of behavior covering a wide range
ofsituationsarepervasivethroughoutthebook.Thisbuildsaknowl-
edgebasethatapractitionercanusetoanticipatetherangeofpossible
resultsthatmaybeencounteredwithanewstructure.
• Theapproachhasstrongvisualcomponents,especiallyintheempha-
sis on consistent semi-quantitative sketching of deformed structures.
Thesesketchesareespeciallyusefulasrepositoriesandenhancements
to experience. The reason is that: i) There is a synergy between such
sketchesandmomentdiagramswhichareessentialfordesignsothat
experience in one translates into improvements in the other. ii) Both
deeper insight and more experience with the analysis of beam and
frame structures allows the user to be more accurate or to add more
details in the sketches of the deformed shapes. Having drawn such
improved sketches, the user then remembers and consolidates both
insight and experience. iii) Comparisons between the sketches of the
deformed structures and moment diagrams allows inconsistencies to
bedetectedandhence,reducespotentialmanualerrors.
• The approachgenerally localizes alldimensional quantities inone or
afewfactorssothatthemainparameterstobeestimatedaretherel-
evant relative stiffnesses. This non-dimensionalization also generally
leads to having all calculated non-dimensional quantities lying be-
tweennegativeandpositiveone.Bothoftheseeffectsreducethelike-
lihood of manual error because the range of possible values become
ix